Lagrange mesh and exact diagonalization for numerical study of semiconductor quantum dot systems with application in singlet-triplet qubits

We present a highly flexible computational scheme for studying correlated electrons confined by an arbitrary external potential in two-dimensional semiconductor quantum dots. The method starts by a Lagrange mesh calculation for the single-particle states, followed by the calculation of the Coulomb interaction matrix elements between these, and combining both in the exact diagonalization of the many-body Hamiltonian. We apply the method in simulation of double quantum dot singlet-triplet qubits. We simulate the full quantum control and dynamics of one singlet-triplet qubit. We also use our method to provide an exact diagonalization based first-principles model for studying two singlet-triplet qubits and their capacitative coupling via the long-distance Coulomb interaction.